Algebra Examples

$f (x) = \frac{- 3 x^{2} + 21 x - 36}{5 x - 20}$

Step 1

Rewrite the function as an equation.

$y = - \frac{3 (x - 3)}{5}$

Step 2

Rewrite in slope-intercept form.

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Step 2.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.2

Write in $y = m x + b$ form.

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Step 2.2.1

Apply the distributive property.

$y = - \frac{3 x + 3 \cdot - 3}{5}$

Step 2.2.2

Multiply $3$ by $- 3$ .

$y = - \frac{3 x - 9}{5}$

Step 2.2.3

Split the fraction $\frac{3 x - 9}{5}$ into two fractions.

$y = - (\frac{3 x}{5} + \frac{- 9}{5})$

Step 2.2.4

Move the negative in front of the fraction.

$y = - (\frac{3 x}{5} - \frac{9}{5})$

Step 2.2.5

Combine the numerators over the common denominator.

$y = - \frac{3 x - 9}{5}$

Step 2.2.6

Factor $3$ out of $3 x - 9$ .

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Step 2.2.6.1

Factor $3$ out of $3 x$ .

$y = - \frac{3 (x) - 9}{5}$

Step 2.2.6.2

Factor $3$ out of $- 9$ .

$y = - \frac{3 x + 3 \cdot - 3}{5}$

Step 2.2.6.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$y = - \frac{3 (x - 3)}{5}$

Step 2.2.7

Reorder terms.

$y = - (\frac{3}{5} (x - 3))$

Step 2.2.8

Remove parentheses.

$y = - \frac{3}{5} (x - 3)$

Step 3

Use the slope-intercept form to find the slope and y-intercept.

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Step 3.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = - \frac{3}{5}$

$b = 0$

Step 3.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $- \frac{3}{5}$

y-intercept: $(0, 0)$

Slope: $- \frac{3}{5}$

y-intercept: $(0, 0)$

Step 4

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

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Step 4.1

Write in $y = m x + b$ form.

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Step 4.1.1

Apply the distributive property.

$y = - \frac{3 x + 3 \cdot - 3}{5}$

Step 4.1.2

Multiply $3$ by $- 3$ .

$y = - \frac{3 x - 9}{5}$

Step 4.1.3

Split the fraction $\frac{3 x - 9}{5}$ into two fractions.

$y = - (\frac{3 x}{5} + \frac{- 9}{5})$

Step 4.1.4

Move the negative in front of the fraction.

$y = - (\frac{3 x}{5} - \frac{9}{5})$

Step 4.1.5

Combine the numerators over the common denominator.

$y = - \frac{3 x - 9}{5}$

Step 4.1.6

Factor $3$ out of $3 x - 9$ .

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Step 4.1.6.1

Factor $3$ out of $3 x$ .

$y = - \frac{3 (x) - 9}{5}$

Step 4.1.6.2

Factor $3$ out of $- 9$ .

$y = - \frac{3 x + 3 \cdot - 3}{5}$

Step 4.1.6.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$y = - \frac{3 (x - 3)}{5}$

Step 4.1.7

Reorder terms.

$y = - (\frac{3}{5} (x - 3))$

Step 4.1.8

Remove parentheses.

$y = - \frac{3}{5} (x - 3)$

Step 4.2

Find the x-intercept.

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Step 4.2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \frac{3}{5} \cdot (x - 3)$

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

Rewrite the equation as $- \frac{3}{5} \cdot (x - 3) = 0$ .

$- \frac{3}{5} \cdot (x - 3) = 0$

Step 4.2.2.2

Multiply both sides of the equation by $- \frac{5}{3}$ .

$- \frac{5}{3} (- \frac{3}{5} \cdot (x - 3)) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3

Simplify both sides of the equation.

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Step 4.2.2.3.1

Simplify the left side.

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Step 4.2.2.3.1.1

Simplify $- \frac{5}{3} (- \frac{3}{5} \cdot (x - 3))$ .

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Step 4.2.2.3.1.1.1

Apply the distributive property.

$- \frac{5}{3} (- \frac{3}{5} x - \frac{3}{5} \cdot - 3) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.2

Combine $x$ and $\frac{3}{5}$ .

$- \frac{5}{3} (- \frac{x \cdot 3}{5} - \frac{3}{5} \cdot - 3) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.3

Multiply $- \frac{3}{5} \cdot - 3$ .

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Step 4.2.2.3.1.1.3.1

Multiply $- 3$ by $- 1$ .

$- \frac{5}{3} (- \frac{x \cdot 3}{5} + 3 (\frac{3}{5})) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.3.2

Combine $3$ and $\frac{3}{5}$ .

$- \frac{5}{3} (- \frac{x \cdot 3}{5} + \frac{3 \cdot 3}{5}) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.3.3

Multiply $3$ by $3$ .

$- \frac{5}{3} (- \frac{x \cdot 3}{5} + \frac{9}{5}) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.4

Move $3$ to the left of $x$ .

$- \frac{5}{3} (- \frac{3 x}{5} + \frac{9}{5}) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5

Simplify terms.

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Step 4.2.2.3.1.1.5.1

Apply the distributive property.

$- \frac{5}{3} (- \frac{3 x}{5}) - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.2

Cancel the common factor of $5$ .

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Step 4.2.2.3.1.1.5.2.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$\frac{- 5}{3} (- \frac{3 x}{5}) - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.2.2

Move the leading negative in $- \frac{3 x}{5}$ into the numerator.

$\frac{- 5}{3} \cdot \frac{- 3 x}{5} - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.2.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{3} \cdot \frac{- 3 x}{5} - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.2.4

Cancel the common factor.

$\frac{5 \cdot - 1}{3} \cdot \frac{- 3 x}{5} - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.2.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x) - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.3

Cancel the common factor of $3$ .

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Step 4.2.2.3.1.1.5.3.1

Factor $3$ out of $- 3 x$ .

$\frac{- 1}{3} (3 (- x)) - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.3.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.3.3

Rewrite the expression.

$- - x - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.4

Multiply.

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Step 4.2.2.3.1.1.5.4.1

Multiply $- 1$ by $- 1$ .

$1 x - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.4.2

Multiply $x$ by $1$ .

$x - \frac{5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.5

Cancel the common factor of $5$ .

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Step 4.2.2.3.1.1.5.5.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$x + \frac{- 5}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.5.2

Factor $5$ out of $- 5$ .

$x + \frac{5 (- 1)}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.5.3

Cancel the common factor.

$x + \frac{5 \cdot - 1}{3} \cdot \frac{9}{5} = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.5.4

Rewrite the expression.

$x + \frac{- 1}{3} \cdot 9 = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.6

Cancel the common factor of $3$ .

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Step 4.2.2.3.1.1.5.6.1

Factor $3$ out of $9$ .

$x + \frac{- 1}{3} \cdot (3 (3)) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.6.2

Cancel the common factor.

$x + \frac{- 1}{3} \cdot (3 \cdot 3) = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.6.3

Rewrite the expression.

$x - 1 \cdot 3 = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.1.1.5.7

Multiply $- 1$ by $3$ .

$x - 3 = - \frac{5}{3} \cdot 0$

Step 4.2.2.3.2

Simplify the right side.

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Step 4.2.2.3.2.1

Multiply $- \frac{5}{3} \cdot 0$ .

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Step 4.2.2.3.2.1.1

Multiply $0$ by $- 1$ .

$x - 3 = 0 (\frac{5}{3})$

Step 4.2.2.3.2.1.2

Multiply $0$ by $\frac{5}{3}$ .

$x - 3 = 0$

Step 4.2.2.4

Add $3$ to both sides of the equation.

$x = 3$

Step 4.2.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0)$

Step 4.3

Find the y-intercept.

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Step 4.3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \frac{3}{5} \cdot ((0) - 3)$

Step 4.3.2

Solve the equation.

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Step 4.3.2.1

Remove parentheses.

$y = - \frac{3}{5} \cdot (0 - 3)$

Step 4.3.2.2

Remove parentheses.

$y = - \frac{3}{5} \cdot ((0) - 3)$

Step 4.3.2.3

Simplify $- \frac{3}{5} \cdot ((0) - 3)$ .

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Step 4.3.2.3.1

Subtract $3$ from $0$ .

$y = - \frac{3}{5} \cdot - 3$

Step 4.3.2.3.2

Multiply $- \frac{3}{5} \cdot - 3$ .

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Step 4.3.2.3.2.1

Multiply $- 3$ by $- 1$ .

$y = 3 (\frac{3}{5})$

Step 4.3.2.3.2.2

Combine $3$ and $\frac{3}{5}$ .

$y = \frac{3 \cdot 3}{5}$

Step 4.3.2.3.2.3

Multiply $3$ by $3$ .

$y = \frac{9}{5}$

Step 4.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{9}{5})$

Step 4.4

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ 0 & \frac{9}{5} \\ 3 & 0 \end{array}$

Step 5

Graph the line using the slope and the y-intercept, or the points.

Slope: $- \frac{3}{5}$

y-intercept: $(0, 0)$

$\begin{array}{c} x & y \\ 0 & \frac{9}{5} \\ 3 & 0 \end{array}$

Step 6